(a) By what factor is one sound more intense than another if the sound has a level 17.0 dB higher than the other?
(b) If one sound has a level 32.0 dB less than another, what is the ratio of their intensities?
a. 10*Log(I1/Io) = 17db
Log(I1/Io) = 1.7
I1/Io = 50.1
b. 10*Log(I1/Io) = -32 db
Log(I1/Io) = -3.2
I1/Io = 0.00063
To calculate the factor by which one sound is more intense than another given a difference in decibel levels, we need to understand the relationship between decibels and sound intensity.
Decibels (dB) are a logarithmic unit used to measure the relative intensity of sound. The formula for calculating the relationship between sound intensity (I) in decibels is:
dB = 10 * log10(I/I₀)
Where I₀ is the reference intensity.
Now let's solve the questions:
(a) By what factor is one sound more intense than another if the sound has a level 17.0 dB higher than the other?
To find the factor, we need to calculate the intensity ratio using the formula:
dB₁ - dB₂ = 10 * log10(I₁/I₂)
Rearranging the equation to solve for the intensity ratio (I₁/I₂):
I₁/I₂ = 10^((dB₁ - dB₂)/10)
Substituting the given values, where dB₁ is 17.0 dB higher than dB₂:
I₁/I₂ = 10^((17.0 dB)/10)
Using the properties of logarithms, we can solve for the intensity ratio:
I₁/I₂ = 10^(1.7) = 50.12
Therefore, one sound is approximately 50.12 times more intense than the other sound.
(b) If one sound has a level 32.0 dB less than another, what is the ratio of their intensities?
Similarly, we can use the same formula:
I₁/I₂ = 10^((dB₁ - dB₂)/10)
Given that dB₁ is 32.0 dB less than dB₂:
I₁/I₂ = 10^((-32.0 dB)/10)
Simplifying the equation:
I₁/I₂ = 10^(-3.2) = 0.005012
Therefore, the ratio of their intensities is approximately 0.005012.
(a) To find the factor by which one sound is more intense than another, we need to calculate the ratio of their intensities. The formula to convert from decibels (dB) to intensity ratio is:
Intensity (in the form of ratio) = 10^(dB/10)
In this case, the sound has a level 17.0 dB higher than the other. So the intensity ratio is:
Intensity ratio = 10^(17.0/10)
Calculating this, we get:
Intensity ratio = 10^(1.7)
Intensity ratio ≈ 50.12
Therefore, one sound is approximately 50.12 times more intense than the other.
(b) Similarly, to find the ratio of intensities when one sound has a level 32.0 dB less than the other, we use the same formula:
Intensity ratio = 10^(dB/10)
In this case, the sound has a level 32.0 dB less than the other. So the intensity ratio is:
Intensity ratio = 10^(-32.0/10)
Calculating this, we get:
Intensity ratio = 10^(-3.2)
Intensity ratio ≈ 0.00501187
Therefore, the ratio of their intensities is approximately 0.00501187.